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A particle
P moves in a straight line with displacement relative to origin given by

s=2sin(pi t)+sin(2pi t),t >= 0,
where
t is the time in seconds and the displacement is measured in centimetres.
(i) Write down the period of the function
s.

A particle $P$ moves in a straight line with displacement relative to origin given by $\newline$ $s=2 \sin (\pi t)+\sin (2 \pi t), t \geq 0,$ $\newline$ where $t$ is the time in seconds and the displacement is measured in centimetres. $\newline$ (i) Write down the period of the function $s$ .

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Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. A particle

P

moves in a straight line with displacement relative to origin given by

\newline

s=2 \sin (\pi t)+\sin (2 \pi t), t \geq 0,

\newline

where

t

is the time in seconds and the displacement is measured in centimetres.

\newline

(i) Write down the period of the function

s

.

Function Components: The function $s$ is a combination of two sine functions, $2\sin(\pi t)$ and $\sin(2\pi t)$ .
Period of Sine Function: The period of a sine function $\sin(Bt)$ is given by $\frac{2\pi}{B}$ .
Period of $2\sin(\pi t)$ : For the first part, $2\sin(\pi t)$ , $B$ is $\pi$ . So, the period is $(2\pi)/\pi = 2$ seconds.
Period of $\sin(2\pi t)$ : For the second part, $\sin(2\pi t)$ , $B$ is $2\pi$ . So, the period is $(2\pi)/(2\pi) = 1$ second.
Period of Function $s$ : The period of the function $s$ is the least common multiple (LCM) of the periods of its components.
Least Common Multiple: The LCM of $2$ and $1$ is $2$ .

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